7.1. Implications for the underlying 3-dimensional structure
The result from Sect. 5, namely that an ensemble of randomly positioned clumps with a given power law mass spectrum and a given mass-size relation has a fractional Brownian motion projected image, brings up the question whether the reverse conclusion can also be drawn: i.e. does the decomposition of an fBm -structure (power law power spectrum and random phases) into (Gaussian) clumps give the corresponding power law mass spectrum? Trying to answer this question clearly touches on two important issues: the first one concerns the connection between the underlying 3-dim structure corresponding to the observed 2-dim projection, the second one concerns the velocity structure related to the fBm -density distribution. It is obvious, that both issues are not independent: only the velocity structure in observed molecular cloud spectra allows the identification of individual clumps, which otherwise would merge into indistinguishable larger structures due to the overlap along the line of sight.
Nevertheless we can derive certain constraints for the 3-dim structure from the fact that the observed 2-dimensional projection has fBm -structure. The following always assumes that the observations are in an optically thin line, so that the observed image directly corresponds to the column density, i.e. the density structure integrated along the line of sight. From the fact that the projection of the 3-dimensional structure, i.e. the integration along the line of sight direction z, in Fourier space corresponds to taking the -cut of the 3-dimensional Fourier transform , we can infer that the 3-dimensional Fourier transform must be such, that its -cut has a power law power spectrum with the same spectral index as its 2-dimensional projection, i.e. the observed image. From the Copernican principle that the line of sight direction cannot be a preferred direction for the cloud structure, we can then conclude that the 3-dimensional cloud density structure must be such, that in Fourier space any 2-dimensional cut through its origin must give a power law power spectrum . This shows that the full 3-dimensional power spectrum also follows a power law, with the same spectral index as that of the 2-dimensional projected image. It does tell nothing, however, about the 3-dimensional phase distribution, although the assumption seems reasonable that they are as randomly distributed as the 2-dimensional phases. This is an ad hoc assumption, however, and one should keep in mind that the phases of the 3-dimensional Fourier image might well have some special correlation that is not visible in the 2-dimensional projection. One should remember that, as was discussed in Sect. 3, the reverse conclusion is always true: the -dimensional projection of an E-dimensional structure with a power law power spectrum has again a power law power spectrum with the same spectral index.
Along this line of reasoning, the 3-dimensional structure thus also has a power law power spectrum whose spectral index is, according to the analysis of the observed 2-dimensional projection, close to 2.8. If the 3-dimensional phase distribution is in fact also completely random, this has very important consequences. It implies that the 3-dimensional structure is completely dominated by surface. A 3-dimensional fBm structure is limited to the range , corresponding to . The corresponding volume-surface relation for the fractal structure of the iso- density surface (the 3-dimensional analogon to the perimeter/area relation of the iso-intensity contours), then has a fractal dimension in the range , i.e. . At the minimum value of , i.e. , this already corresponds to the surface increasing proportional to the volume. The value of for a 3-dim structure derived above, would nominally imply , i.e. an even faster increase of surface with volume.
For a full understanding of the 3-dim cloud structure a link to the velocity structure is crucial. The velocity structure is important not only as a tool to tell overlapping 3-dim spatial structure apart, and to keep the emission of individual clumps from mutual shielding in the radiative transfer. Physically, the velocity structure must be linked to the density structure via the magneto-hydrodynamic equations describing the turbulent internal cloud motions. Ultimately, a proper description of this magneto hydrodynamic turbulence should describe the cloud structures observed. We are obviously far from reaching this goal. A simple approach, assuming independent fBm -distributions both for the density and the velocity structure, results in surprisingly realistic looking molecular cloud spectra which even satisfy some scaling relations such as the size-line width relation, , over a limited range of scales. Future work will have to investigate these aspects in detail. Observationally we can get further constraints on the velocity structure e.g. by applying the -variance analysis introduced above not only to the integrated intensity image of the molecular cloud, but also to individual velocity channel maps. These investigations will be presented in a future paper together with the -variance analysis of a broader selection of observed molecular clouds (Bensch et al., in prep. ).
7.2. Observational limits
The fact that detailed observations of molecular cloud structure, covering a significant dynamic range in spatial scales and also in the signal to noise ratio within a reasonable integration time, have become possible only within the last decade is linked to rapid progress in technology. The high sensitivity of large mm-wave telescopes and the excellent low noise performance of SIS-heterodyne receivers available at the low-J carbon monoxide line frequencies result in acceptable observing times of order several days for a decent size cloud mapping project. However, higher sensitivity receivers are not to be expected as the present day receivers already reach close to quantum limited performance; also, telescopes substantially larger than the IRAM 30m-telescope or the planned 50m LMT project are beyond technical feasibility. Mapping speed will profit from future array receiver systems. Extension of the structure analysis down to angular scales below the diffraction limit of the large single dish telescopes will only be possible with interferometric techniques.
It is thus of particular importance to estimate the observing time needed for extension of a cloud map to higher angular resolution with an interferometer array. Let us thus consider for simplicity and consistency the situation, where we want to observe a single resolution element of a large scale cloud map with higher angular resolution using an interferometer of antennas identical to the single dish used for the low angular resolution map. The effective system temperature (including receiver sensitivity, telescope efficiency and atmospheric losses) for each interferometer element and the single dish telescope are assumed to be identical. The noise level reachable in a given integration time and resolution bandwidth with the interferometer scales as (see Downes 1989)
where is the noise achieved with the single dish telescope in the same given integration time, L is the length of the longest baseline, D is the diameter of each interferometer antenna, and is the number of baselines for a number n of interferometer elements. The resolution achieved with the interferometer scales as . The noise per resolution element then scales as
The interferometer observes all resolution elements within its field of view simultaneously. The single dish telescope has to observe each resolution element separately for a time . The scaled up single dish map with the same number of pixels as the interferometer map thus takes a total time . For the interferometer map to be a true scaled down version of the larger map, the signal to noise ratio per resolution element has to be the same as for the single dish map. The scaling of the signal level with resolution is given by the assumed fractional Brownian motion structure, which says that the structure in the signal at a scale scales with (see Sect. 3). The signal structure expected at both resolutions thus scales as
Combining the above relations and solving for equal signal to noise at the resolution of both maps gives the scaling of the total integration times:
The same equation can be derived by considering the increase in integration time with higher resolution according to due to the decrease of the signal level for an otherwise identical, but larger single dish telescope; the reduced efficiency of an n-element interferometer of the same total size (and hence angular resolution) due to only a fraction of the aperture being filled, this efficiency entering squared into the ratio of integration times; and the spatial multiplexing advantage of the interferometer, , observing all resolution elements within the field of view simultaneously. Combining these factors and using reproduces the above relation.
We see that the total integration time rapidly increases with increasing resolution, . This increase is compensated by the decrease of integration time in proportion to the number of interferometer baselines. With the value of derived above, and assuming a 5-element interferometer such as the IRAM Plateau-de-Bure instrument, i.e. , a submap of a single resolution element of the large scale single dish map with the interferometer at a 10 times higher resolution would thus take about 60 times as long as the total large scale map. This is unrealistically long to persue, considering the already very long observing times needed for decent size single dish maps with adequate signal to noise ratio. Extending studies of cloud structures to higher angular resolution will thus be feasible only with very large () future interferometer arrays.
7.3. Beyond fractional Brownian motion structure
The present paper discusses molecular cloud structure in the framework of what is commonly called "monofractal structure". The actual cloud structure is certainly more complex and many different structures with the same mono-fractal characteristics are possible. A few recent papers have attempted to characterise observed clouds with multifractal methods. They show that the cloud structure indeed shows multifractal properties, both in the velocity distribution (Miesch & Bally 1994) and in the column density distribution (Chappell & Scalo 1997). However, they also seem to indicate that these multifractal properties vary from cloud to cloud. Whether this is due to the limited data base available and possible systematic errors in the analysis, or whether observed molecular clouds in fact fall into a single "universality class" is not clear at the moment. Molecular clouds clearly need more than one single parameter, e.g. a single fractal dimension, a single power spectrum spectral index, or a single power law index of their clump mass distribution, to fully characterise their structure. In the present paper, we explicitely excluded multifractal aspects from the present discussion and rather concentrate on a comparison of and the possibility to unify the various monofractal measures of cloud structure obtained with the different methods commonly used.
© European Southern Observatory (ESO) 1998
Online publication: July 20, 1998